PREFACE. xi 



In the second chapter, we apply this principle of conserva- 

 tion of probability of phase to the theory of errors in the 

 calculated phases of a system, when the determination of the 

 arbitrary constants of the integral equations are subject to 

 error. In this application, we do not go beyond the usual 

 approximations. In other words, we combine the principle 

 of conservation of probability of phase, which is exact, with 

 those approximate relations, which it is customary to assume 

 in the " theory of errors." 



In the third chapter we apply the principle of conservation 

 of extension-in-phase to the integration of the differential 

 equations of motion. This gives Jacobi's " last multiplier," 

 as has been shown by Boltzmann. 



In the fourth and following chapters we return to the con- 

 sideration of statistical equilibrium, and confine our attention 

 to conservative systems. We consider especially ensembles 

 of systems in which the index (or logarithm) of probability of 

 phase is a linear function of the energy. This distribution, 

 on account of its unique importance in the theory of statisti- 

 cal equilibrium, I have ventured to call canonical, and the 

 divisor of the energy, the modulus of distribution. The 

 moduli of ensembles have properties analogous to temperature, 

 in that equality of the moduli is a condition of equilibrium 

 with respect to exchange of energy, when such exchange is 

 made possible. 



We find a differential equation relating to average values 

 in the ensemble which is identical in form with the funda- 

 mental differential equation of thermodynamics, the average 

 index of probability of phase, with change of sign, correspond- 

 ing to entropy, and the modulus to temperature. 



For the average square of the anomalies of the energy, we 

 find an expression which vanishes in comparison with the 

 square of the average energy, when the number of degrees 

 of freedom is indefinitely increased. An ensemble of systems 

 in which the number of degrees of freedom is of the same 

 order of magnitude as the number of molecules in the bodies 



